Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data

Feldman, M.; Tudorascu, A.

doi:10.1090/spmj/1403

Lagrangian solutions for the semi-geostrophic shallow water system in physical space with general initial data
HTML articles powered by AMS MathViewer

by M. Feldman and A. Tudorascu

St. Petersburg Math. J. 27 (2016), 547-568

DOI: https://doi.org/10.1090/spmj/1403

Published electronically: March 30, 2016

PDF | Request permission

Abstract:

In order to accommodate general initial data, an appropriately relaxed notion of renormalized Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space is introduced. This is shown to be consistent with previous notions, generalizing them. A weak stability result is obtained first, followed by a general existence result whose proof employs the said stability and approximating solutions with regular initial data. The renormalization property ensures the return from physical to dual space and ultimately enables us to achieve the desired results.

References

Luigi Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math. 158 (2004), no. 2, 227–260. MR 2096794, DOI 10.1007/s00222-004-0367-2
Luigi Ambrosio, Maria Colombo, Guido De Philippis, and Alessio Figalli, Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case, Comm. Partial Differential Equations 37 (2012), no. 12, 2209–2227. MR 3005541, DOI 10.1080/03605302.2012.669443
—, A global existence result for the semigeostrophic equations in three dimensional convex domains, Preprint, arXiv:1205.5435, 2012.
Luigi Ambrosio and Wilfred Gangbo, Hamiltonian ODEs in the Wasserstein space of probability measures, Comm. Pure Appl. Math. 61 (2008), no. 1, 18–53. MR 2361303, DOI 10.1002/cpa.20188
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498
J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem, SIAM J. Appl. Math. 58 (1998), no. 5, 1450–1461. MR 1627555, DOI 10.1137/S0036139995294111
Yann Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. MR 1100809, DOI 10.1002/cpa.3160440402
M. Cullen, Private communication.
Michael Cullen and Mikhail Feldman, Lagrangian solutions of semigeostrophic equations in physical space, SIAM J. Math. Anal. 37 (2006), no. 5, 1371–1395. MR 2215268, DOI 10.1137/040615444
Mike Cullen and Wilfrid Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Ration. Mech. Anal. 156 (2001), no. 3, 241–273. MR 1816477, DOI 10.1007/s002050000124
Mike Cullen and Hamed Maroofi, The fully compressible semi-geostrophic system from meteorology, Arch. Ration. Mech. Anal. 167 (2003), no. 4, 309–336. MR 1981860, DOI 10.1007/s00205-003-0245-x
M. J. P. Cullen and R. J. Purser, An extended Lagrangian theory of semigeostrophic frontogenesis, J. Atmospheric Sci. 41 (1984), no. 9, 1477–1497. MR 881109, DOI 10.1175/1520-0469(1984)041<1477:AELTOS>2.0.CO;2
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Josiane C. O. Faria, Milton C. Lopes Filho, and Helena J. Nussenzveig Lopes, Weak stability of Lagrangian solutions to the semigeostrophic equations, Nonlinearity 22 (2009), no. 10, 2521–2539. MR 2539766, DOI 10.1088/0951-7715/22/10/011
Mikhail Feldman and Adrian Tudorascu, On Lagrangian solutions for the semi-geostrophic system with singular initial data, SIAM J. Math. Anal. 45 (2013), no. 3, 1616–1640. MR 3061466, DOI 10.1137/120870116
—, On the semigeostrophic system in physical space with general initial data. (to appear)
W. Gangbo, T. Nguyen, and A. Tudorascu, Euler-Poisson systems as action-minimizing paths in the Wasserstein space, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 419–452. MR 2505360, DOI 10.1007/s00205-008-0148-y
B. Hoskins, The geostrophic momentum approximation and the semigeostrophic equations, J. Atmospheric. Sci. 32 (1975), no. 2, 233–242.
Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058